There is a real-valued random variable $R$. Define a finite set of random variables ("data points") $$X_i = R + Z_i \; \text{for } i\in\{1,\ldots,n\},$$ where $Z_i$ are identically and independently distributed, mean-zero, and independent of $R$. The prior distributions of $R$ and the $Z_i$ are given; the support of each is the real line.

Define the conditional, or posterior, expectation of $R$ given particular realizations of the data $$\hat{R}(x_1,\ldots,x_n)= \mathbb{E}[R \mid X_1=x_1,\ldots,X_n=x_n]. $$ (This is the expectation of $R$ assessed by someone who sees the data points but not $R$ itself, and is a measurable function $\mathbb{R}^n \to \mathbb{R}$.) The question is: how much can an adversary with a given amount of manipulation power over a single data point move this estimate?

More precisely, fix a number $\Delta$. I am looking for a bound, which is useful as $n$ grows large, on $$M_n(\Delta)=\mathbb{E}[\hat{R}(X_1+\Delta,X_2,\ldots,X_n) - \hat{R}(X_1,X_2,\ldots,X_n)].$$ This expectation is an integral over all uncertainty in the model (i.e. in $R$ and the $X_i$), though I believe it should be possible to give a good bound even conditional on $R$.

We can take all random variables to be square-integrable if necessary, and make any other convenient assumptions.

A conjecture is that the manipulability is small in the sense that $M(\Delta) \to 0 $ as $n \to \infty$ and indeed $M_n'(\Delta) \to 0$ as well.

The conclusion may seem obvious because the posterior distribution of $R$ conditional on the data $X_1,\ldots,X_n$ converges to a point mass whose location is independent of the realization of $X_1$. But this does not readily imply a bound on the $L^1$ norm of the difference between $\hat{R}(X_1+\Delta,\ldots,X_n)$ and $R$, or the difference between $\hat{R}(X_1+\Delta,\ldots,X_n)$ and the unmanipulated estimate $\hat{R}(X_1,\ldots,X_n)$. It could be that the manipulation has a slowly decaying probability of achieving very large deviations in the estimate, so that it messes up the expectation.

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